Sharpe introduced his reward per unit of variability ratio as the difference between the mean return on a portfolio ( rp) and the mean risk free rate ( rf) divided by the standard deviation of the portfolio's excess return over the risk free rate, denoted σxp, as shown in equation (1):
                              S          p                =                                                            r                -                            p                        -                                          r                f                            -                                            σ            xp                                              (        1        )            
The Sharpe ratio is recommended by Capital Asset Pricing Theory and theoretical writings on investment portfolio performance evaluation as the best measure for the evaluation of the risk adjusted investment performance of an investor's entire portfolio. Hence, it can fairly be argued to represents one of the best investment performance measure for the average individual investor.
Despite being considered by many as the most popular investment performance measure, the Sharpe ratio carries a major reliability burden. This is because when an asset experiences a negative average excess return, the Sharpe ratio indicates better investor utility ratings the riskier the asset. Many previous studies lack reliability, as they assess negative average excess return assets using the Sharpe ratio without addressing or even acknowledging this problem.
If the Sharpe ratio is used to assess the utility of an asset with negative average excess return, it considers the asset the better the higher its risk. For instance, comparing assets A and B with the average excess returns −5% and −8% as well as the (excess return) standard deviations 10% and 20%, respectively, asset B with the bigger loss and the much higher risk receives the better Sharpe ratio (−0.4 compared to −0.5).
FIG. 1 is a modification of Sharpe's (1998: 31) FIG. 10 titled ‘Performance of Two Funds in Bad Times”. During times, in which the market return is smaller than the risk free return (rf), it compares the utility resulting for investors from different investment strategies indicated by the points, whereby the Sharpe ratio ascribes all points Y the same utility and the same excess utility over all points X's utility. Hence Sharpe, who only displays the black points, considers any investment Y to be more attractive than any investment X. However, based on the basic axioms that the measurement of investor utility is the original function of any investment performance evaluation measure and that risk has a negative utility, it can be shown that the Sharpe ratio severely lacks reliability in this case, as it recommends investments with the same return but a higher risk (i.e. standard deviation) than an alternative. For instance, the Share ratio recommends investment Y′ instead of X′, Y″ instead of X, and Y′″ instead of X″ in FIG. 1.
FIG. 2 illustrates the negative Sharpe ratio dilemma for an investment with a −100% average excess return and various excess return standard deviation values ranging from 25% to 200%. These high excess return standard deviation values have been chosen to show the effect of marginal increases in excess return standard deviation for both sides of 1 (100%). The return of 100% has been selected to receive a Sharpe ratio of 1 for an excess return standard deviation of 1. As shown by the graph, the Sharpe ratio generally assesses an investment with equal loss but a higher risk with a better, as less negative, investor utility score.
But this negative average excess return problem does not only apply to the Sharpe ratio, it also applies in some form to any measure, which is based on the ratio of return to risk (e.g. Sortino Ratio (Sortino, F. A. and R. van der Meer. (1991) “Downside risk.” Journal of Portfolio Management, 17 (4): 27-31) or Modified Sharpe ratio (Gregoriou, G. N. and J.-P. Gueyie. (2003) “Risk-Adjusted Performance of Funds of Hedge Funds using a Modified Sharpe Ratio.” Journal of Wealth Management, Winter 2003: 77-83)). The reliability problem of the Sharpe ratio is even greater, if a study analyzes the mean of many funds' Sharpe ratios, since any inclusion of meaningless negative Sharpe ratios into an average Sharpe ratio makes this mean itself over-proportionally more meaningless.
One of several proposals for overcoming the problem with the Sharpe ratio has been suggested by Israelsen. This is described in “Sharpening the Sharpe Ratio.” Financial Planning, 33 (1): 49-51, Israelsen (2003) and “A refinement to the Sharpe ratio and information ratio.” Journal of Asset Management, 5 (6): 423-427, Israelsen, C. L. (2005). This proposal involves multiplying a portfolio's average excess return with its standard deviation, if the former is negative, as shown in equation (2):
                              I          p                =                                            r              -                        xp                                σ            xp                          (                                                                    r                    -                                    xp                                /                                                                                              r                      -                                        xp                                                                                )                                                          (        2        )            where Ip denotes the Israelsen ratio; rxp is the mean excess return of the portfolio over the risk free rate; | rxp| is the same mean excess return in absolute terms; and σxp is the standard deviation of the portfolio's excess return. However, a problem with the Israelsen ratio is that it leads to unreliable fund ratings. If the average excess return standard deviation in a sample of funds is far above one, positive Israelsen ratios experience low absolute values, but negative Israelsen ratios show big absolute values. This bias, which Israelsen himself appears to recognize, prohibits the calculation of any reliable average including positive and negative Israelsen ratios.
All other current attempts to solve the negative average excess return problem of the Sharpe ratio or a similar ratio result in investor utility ratings, which themselves face even greater reliability problems than the Israelsen ratio. Hence, no reliable return to risk ratio for the assessment of loss incurring assets appears to currently exist. Given that investors are expected to be especially risk averse, when their investments incur losses and that return to risk ratios are the main means of assessing investors' entire portfolios, this lack of a reliable measure is a problem.
In summary, the Sharpe ratio loses its meaning in case of negative average excess return and hence can be considered an inappropriate measure of investor utility unless all observed relevant average excess returns are positive. Since the negative average excess return problem applies to the Sharpe ratio as well as its common substitutes, investors, fund managers and other financial market participants have a clear, strong need for a Sharpe ratio like measure of investor utility which they can appropriately use for assets with positive or negative average excess return. This need is especially strong for big institutional investors, as they face substantial negative price impacts when attempting to offset a large portion of an asset due to the expectation of negative excess returns.